1 Introduction
Conjugacy classes of finite Coxeter groups have long been of
interest, the correspondence between partitions and conjugacy
classes of the symmetric groups having been observed by Cauchy
[4] in the early days of group theory. For Coxeter groups
of type Bn and Dn, descriptions of their conjugacy classes, by
Specht [12] and Young [13], have also been known
for a long time. In 1972, Carter [2] gave a uniform and
systematic treatment of the conjugacy classes of Weyl groups. More
recently, Geck and Pfeiffer [6] reworked Carter’s
descriptions from more of an algorithmic standpoint. Motivation for
investigating the conjugacy classes of finite Coxeter groups, and
principally those of the irreducible finite Coxeter groups, has come
from many directions, for example in the representation theory of
these groups and the classification of maximal tori in groups of Lie
type (see [3]). The behaviour of length in a conjugacy
class is frequently important. Of particular interest are those
elements of minimal and maximal lengths in their class. Instrumental to Carter’s work was establishing the fact that in a finite Coxeter group every element is either an involution or a product of two involutions. Given the importance of the length function, it is natural to ask whether for an element w it is possible to choose two involutions σ and τ with w=στ
in such a way that combining a reduced expression for σ with one for τ produces a reduced expression for w. That is, can we ensure that the length ℓ(w) is given by ℓ(w)=ℓ(σ)+ℓ(τ)? Not surprisingly, the answer to this is, in general, no. This naturally leads to introducing the concept of the excess of w, denoted by e(w), and defined by
e(w)=min{ℓ(σ)+ℓ(τ)-ℓ(w):στ=w,σ2=τ2=1}.
In [7], [8] and [9], various properties of excess were investigated. It was shown, among other things, that in every conjugacy class of a Coxeter group W there is an element of w of minimal length in the conjugacy class, such that the excess of w is zero [8, Theorem 1.1]. This raises the question as to whether there is also an element of maximal length and excess zero.
In this paper we address this question and show that elements of maximal length and excess zero do indeed exist.
Theorem 1.1.
Let W be a finite Coxeter group and C a conjugacy class of W. Then there exists an element w of maximal length in C such that e(w)=0.
In the course of proving this result we need a workable description of representatives of maximal length in conjugacy classes of Coxeter groups of types An, Bn and Dn. Minimal
length elements in conjugacy classes of Coxeter groups have received considerable attention – see
[6]. Now every
finite Coxeter group W possesses a (unique) element
w0 of maximal length in W. For C a conjugacy class of W,
set C0=Cw0={ww0:w∈C}. If, as happens in many
cases, w0∈Z(W), then C0 is also a conjugacy class of W. Moreover, w∈C has minimal length in C if and only if
ww0 has maximal length in C0. Thus information about
maximal length elements in a conjugacy class may be obtained from
that known about minimal length elements. Among the finite
irreducible Coxeter groups, only those of type Im (m odd),
An, Dn (n odd) and E6 have w0∉Z(W).
The first of these, being just dihedral groups, are quickly dealt
with. Descriptions of maximal length elements in conjugacy classes
of type An were given by Kim [11] and for E6 see [5, Table
III]. In Section 3 of this paper we deal with type Dn (and in doing so give a result for type Bn at the same time). Representatives of maximal length
for type Dn could be extracted from [5, Section 4], but here we give a more direct treatment that deals with both type Bn and type Dn and gives more information about the number of long and short roots taken negative by elements of maximal length. Theorem 3.1 gives an expression for the maximal length of elements in a given conjugacy class for type Dn while Theorem 1.2 below gives a list of maximal length class representatives in types Bn and Dn, and this is what we require for our work on elements of excess zero.
Theorems 1.2 and 3.1 are consequences of a more general result, Theorem 3.6, concerning D-lengths and B-lengths of elements in a Coxeter group W of type Bn (D-length and B-length will be defined in Section 3). Suppose W^ is of type Dn. Then we may regard W^ as a canonical index 2 subgroup of W, where W is a Coxeter group of type Bn. Let C be a conjugacy class of W that is contained in W^. In the case when n is odd, w0≠w^0 (the longest element of W^) and consequently C0=Cw0 is not even a subset of W^, much less a conjugacy class of W^. However, working in the wider context of W, we are able
to obtain elements of maximal D-length in C from suitable elements of minimal B-length in C0. Therefore, in the course of establishing Theorem 3.1, we also produce representative elements of maximal length in their conjugacy class. To describe these elements, we will take for our group of type Bn the group of signed permutations; that is, permutations w of {1,…,n,-1,…,-n} such that w(-i)=-w(i) for 1≤i≤n. Signed permutations can be written as permutations, where each number has either a plus or a minus sign above it. So, for example, if w=(1+2-3-), then w(1)=2, w(-1)=-2, w(2)=-3, w(-2)=3, w(3)=-1 and w(-3)=1. The set of signed permutations where an even number of minus signs appear is a subgroup which is of type Dn. Conjugacy classes in types Bn and Dn are parameterized by signed cycle type (this will be described fully in Section 3), with some classes splitting in type Dn.
For n a natural number, an ordered sequence λ=(λ1,…,λm) with λ1+…+λm=n is called a composition of n. A partition of n is a composition of n, λ=(λ1,…,λm), with λ1≥λ2≥…≥λm. So as there is no confusion between compositions and permutations, for cycles we do not use commas but space out the elements of the cycle. Thus, for example, (1,3,2) is a composition of 6 while (1 3 2) is a permutation in Sym(3). Now,
let λ=(λ1,…,λm) be a composition of n, and let ρ≥0. For ease of notation set μi=∑j=1i-1λj (and by convention μ1=0).
We then define the corresponding signed elementwλ,ρ to be
wλ,ρ=w1⋯wm, where
wi={(μi+1-μi+2-⋯μi+1-1-μi+1-)if 1≤i≤ρ,(μi+1-μi+2-⋯μi+1-1-μi+1+)if ρ<i≤m.
We call λ a maximal split partition (with respect to ρ) if λ1≥⋯≥λρ and λρ+1≥⋯≥λm.
For example, if λ=(5,2,4,3) and ρ=2, then
wλ,ρ=(1-2-3-4-5-)(6-7-)(8-9-10-11+)(12-13-14+).
Our second main result in this paper is the following.
Theorem 1.2.
Let W be of type Bn and W^ its canonical subgroup of type Dn.
Every conjugacy class of W contains an element wλ,ρ, where λ is a maximal split partition with respect to ρ. Each element wλ,ρ has maximal B-length and maximal D-length in its conjugacy class of W. Moreover, the excess of wλ,ρ is zero, both with respect to the length function of W and, if wλ,ρ∈W^, with respect to the length function of W^.
Representatives of minimal length in conjugacy classes of types Bn and Dn appear in [6, Theorems 3.4.7 and 3.4.12]. However we need additional information about elements of minimal length in W-conjugacy classes, which gives as a byproduct (in Corollaries 3.4 and 3.5) an alternative proof that the representatives given in [6] are indeed of minimal length.
In the rest of this section we briefly discuss the proof of Theorem 1.1. Given a root system Φ for a Coxeter group W, we have that Φ is the disjoint union of the set of positive roots Φ+ and the set of negative roots Φ-=-Φ+. For detail on root systems, including these observations, see for example [10, Chapter 5]. It is well known (for example [10, Proposition 5.6]) that for any w in W, the length ℓ(w) is given by
ℓ(w)=|N(w)|=|{α∈Φ+:w(α)∈Φ-}|.
That is, ℓ(w) is the number of positive roots taken negative by w. We emphasise here that, in line with other work on Coxeter groups, elements of the group will act on the left. It is easy to show that if w=gh for some g,h∈W, then
(1.1)ℓ(w)=ℓ(g)+ℓ(h)-2|N(g)∩N(h-1)|.
(Equation (1.1) is well known but is stated and proved as part of [8, Lemma 2.1].) Our method of proving Theorem 1.1 for the classical Weyl groups will be as follows. First we will establish a collection of elements w constituting a representative of maximal length for each conjugacy class of the group under consideration. For each such w, we will obtain involutions σ and τ such that N(σ)∩N(τ)=∅ and στ=w. It follows from equation (1.1) that the excess of w is zero.
We conclude this section with two lemmas which will be useful later.
Lemma 1.3.
Let W be a Coxeter group.
Let g,h∈W and suppose N(g)∩N(h-1) is empty. Then N(gh)=∪˙N(h)h-1(N(g)).
Proof.
Note that |N(h)∩h-1N(g)|=|hN(h)∩N(g)|≤|Φ-∩N(g)|=0. So N(h) and h-1(N(g)) are indeed disjoint. Suppose α∈N(h). Then gh(α)∈Φ+ would imply that h(α)∈-N(g), which implies -h(α)∈N(h-1)∩N(g), a contradiction. Hence we have gh(α)∈Φ-, meaning N(h)⊆N(gh). Now suppose α∈N(gh)∖N(h). Then h(α)∈Φ+ but gh(α)∈Φ-. Therefore α∈h-1(N(g)). Conversely, since N(h-1)∩N(g)=∅, it follows that h-1(N(g))⊆Φ+ and so h-1(N(g))⊂N(gh). Therefore N(gh)=∪˙N(h)h-1(N(g)).
∎
Lemma 1.4.
Let W be a Coxeter group.
Suppose t1,t2,…,tm are involutions with the property that whenever i≠j we have ti(N(tj))=N(tj). Then
N(t1⋯tm)=⋃˙i=1mN(ti).
Proof.
The result clearly holds when m=1. Assume the result holds for m=k. Set uk=t1t2⋯tk. Then inductively
N(uk)=⋃˙i=1kN(ti).
If α∈N(uk), then α∈N(ti) for some i≤k and so tk+1(α)∈N(ti)⊆Φ+. Thus N(uk)∩N(tk+1)=∅. Lemma 1.3 now gives
N(uk+1)=∪˙N(tk+1)tk+1(N(uk))=⋃˙i=1k+1N(ti).
The result follows by induction.
∎
Finally, for σ∈Sym(n), the support of σ, denoted supp(σ) is simply the set of points not fixed by σ. That is,
supp(σ)={i∈{1,…,n}:σ(i)≠i}.
We thank the referee for his/her helpful comments.
2 Type An-1
The permutation group Sym(n) is a Coxeter group of type An-1. So throughout this section we will set W=Sym(n). In this context then, the length of an element w is the number of inversions, that is the number of pairs (i,j) with 1≤i<j≤n such that w(i)>w(j). We can also think of this in terms of the root system (which we can consider as a warm up for the type Bn and Dn cases). For the root system Φ we can take
Φ+={ei-ej:1≤i<j≤n}
and Φ-=-Φ+. Hence
N(w)={ei-ej:i<j,w(i)>w(j)}.
For what follows it will sometimes be helpful to consider intervals [a,b], 1≤a<b≤n. The group Sym([a,b]) is a standard parabolic subgroup of W, and by Φ[a,b]+ we mean {ei-ej:a≤i<j≤b}. We note that if w∈Sym([a,b]), then we have N(w)⊆Φ[a,b]+. The conjugacy classes of W are parameterized by partitions of n. Kim [11] has described a set of representative elements of maximal length in conjugacy classes of Sym(n), using the “stair form”. Following [11], we give the following definition.
Definition 2.1.
Let n be a positive integer.
Define the sequence a1,a2,…,an by a2i-1=i and a2i=n-(i-1). (So a1=1, a2=n, a3=2, a4=n-1 and so on.)
Given a composition λ=(λ1,…,λm) of n, its corresponding element is the element of Sym(n) defined by
wλ=w1w2⋯wm,
where wi=(aλ1+⋯+λi-1+1aλ1+⋯+λi-1+2⋯aλ1+⋯+λi-1+λi).
Let λ=(λ1,…,λm) be a composition of n. Then λ is a maximal composition of n if there exists ℓ, with 0≤ℓ≤m such that λ1,…,λℓ are even numbers in any order, and λℓ+1,…,λm are odd numbers in decreasing order.
For example, given the maximal composition (4,5) of 9, the corresponding element is (1 9 2 8)(3 7 4 6 5).
Any partition of n can be reordered so as to produce a maximal composition. Therefore each conjugacy class can be represented by a maximal composition. We can now state the main result of [11].
Theorem 2.2 (Kim [11]).
Let λ=(λ1,…,λm) be a maximal composition of n. The corresponding element wλ of λ has maximal length in its conjugacy class.
Given a sequence b1,b2,…,bk, of distinct elements in {1,…,n}, we define gb1,…,bk to be the permutation that reverses the sequence and fixes all other elements c∈{1,…,n}, so that g(bi)=bk+1-i. That is,
g=(b1bk)(b2bk-1)⋯(b⌊k/2⌋b⌈k/2⌉+1).
In particular, gb1,…,bk is an involution.
Let w be the k-cycle (b1b2⋯bk). Define
(2.1)σ(w)={gb1,…,bkif k even,gb2,…,bkif k odd,
(2.2)τ(w)={gb1,…,bk-1 if k even,gb1,…,bk if k odd.
Lemma 2.3.
Let w be a cycle of Sym(n). Then writing σ=σ(w) and τ=τ(w) we have that w=στ, where σ and τ are both involutions.
Proof.
It is clear from the definitions that σ and τ are involutions. Let
w=(b1⋯bk).
If k is even, then by (2.1) and (2.2), we see that for i≤k-1 we have
στ(bi)=σ(bk-i)=b(k+1)-(k-i)=bi+1,
and στ(bk)=σ(bk)=b1. Therefore w=στ. If k is odd, then σ(bj)=bk+2-j when 2≤j≤k, and σ(b1)=b1. Therefore, when i≤k-1, we have
στ(bi)=σ(bk+1-i)=bk+2-(k+1-i)=bi+1
and στ(bk)=σ(b1)=b1. Again we get w=στ.
∎
Before we go further we introduce some additional notation. Any composition λ (via its corresponding element wλ) induces a partition X=(X1,…,Xm) of {1,…,⌈n2⌉} and a partition Y=(Y1,…,Ym) of {⌈n2⌉+1,…,n} by setting
Xk={1,…,⌈n2⌉}∩supp(wk),
Yk={⌈n2⌉+1,…,n}∩supp(wk).
By the definition of wk we see that Xk is an interval [x¯k,x¯k], where x¯k and x¯k are, respectively, the minimal and maximal elements of Xk appearing in supp(wk). Similarly we may write Yk=[y¯k,y¯k] for appropriate y¯k and y¯k. For example, if λ=(8,5), then
wλ=(1 13 2 12 3 11 4 10)(5 9 6 8 7)
and we have
X1={1,2,3,4}=[1,4],X2={5,6,7},
Y1={10,11,12,13},Y2={8,9}.
Note also that
σ(w1)=(1 10)(2 11)(3 12)(4 13),σ(w2)=(6 8)(7 9),
τ(w1)=(1 4)(2 3)(11 13),τ(w2)=(5 7)(8 9).
We will see that τ(wk) leaves the sets Xk and Yk invariant, and σ(wk) interchanges, in an order-preserving way, nearly all, if not all, elements of Xk and Yk.
Proposition 2.4.
Let λ be a maximal composition of n and let C be the corresponding conjugacy class of Sym(n). Then the corresponding element wλ has excess zero.
Proof.
Write λ=(λ1,…,λm).
Set w=wλ=w1⋯wm, where wi is as given in Definition 2.1. For each i set σi=σ(wi) and τi=τ(wi). Since the supports (in other words the sets of non-fixed points) of σ and τ are subsets of the support of wi, it is clear that both σi and τi commute with both σj and τj whenever i≠j. Hence σ=σ1⋯σm and τ=τ1⋯τm are involutions with the property that στ=w.
We must show that N(σ)∩N(τ)=∅. This will imply by equation (1.1) that e(w)=0.
Consider the cycle wk of w. Then wk=(aL+1aL+2⋯aL+λk) (setting L=∑j=1k-1λj). This means, depending on the parity of L, that there is some i≥1 for which wk is either (in-i+1i+1n-i⋯) or (n-i+2in-i+1⋯). The support of wk is Xk∪Yk.
Let us consider τk=τ(wk). Now
τk={∏i=1λk2-1(aL+iaL+λk-i)if λk is even,∏i=1⌊λk2⌋(aL+iaL+λk+1-i)if λk is odd.
In both cases τk is mapping odd terms of the sequence (ai) to odd terms and even terms to even terms. In particular, τk∈Sym(Xk)×Sym(Yk). Therefore
(2.3)N(τk)⊆{ei-ej:x¯k≤i<j≤x¯k}∪{ei-ej:y¯k≤i<j≤y¯k}.
(If λk is odd, then we have equality here and τk=gx¯k,…,x¯kgy¯k,…,y¯k.)
Next we look at σk. Setting μ=⌊λk2⌋, we have
σk={∏i=1μ(aL+iaL+λk+1-i)if λk is even,∏i=2μ+1(aL+iaL+λk+2-i)if λk is odd.
What happens this time is that σk is the order preserving bijection between the highest μ elements of Xk and the lowest μ elements of Yk.
Therefore
(2.4)N(σk)={ei-ej:x¯k≤x¯k+1-μ≤i≤x¯k<j<y¯k}∪{ei-ej:x¯k<i<y¯k≤j≤y¯k+μ-1≤y¯k}.
Now for ℓ≠k, we have that σℓ fixes all i for i∉Xℓ∪Yℓ and interchanges various elements of Xℓ and Yℓ. Therefore σℓ(N(σk))=N(σk). So we may apply Lemma 1.4 to conclude that N(σ)=⋃˙Nk=1m(σk). Similarly since τℓ fixes all i for i∉Xℓ∪Yℓ, we can deduce that N(τ)=⋃˙Nk=1m(τk). Looking at equations (2.3) and (2.4), it is clear that N(τ)∩N(σ)=∅. Therefore by equation (1.1) we see that ℓ(w)=ℓ(σ)+ℓ(τ) and hence e(w)=0, as required.
∎
We remark that Theorem 1.1 for type An follows immediately from Theorem 2.2 and Proposition 2.4.
3 Maximal lengths in types Bn and Dn
Throughout this section, W is assumed to be a Coxeter group of
type Bn containing W^, the canonical index 2 subgroup of
type Dn. We will view elements of W as signed cycles.
A cycle is called negative if it has an odd number of minus signs above its entries, and positive otherwise.
The conjugacy classes of W are parameterized by
signed cycle type. So for X a subset of a conjugacy class of
W, this data may be encoded by
λ(X)=(λ1,…,λνX;λνX+1,…,λzX),
where νX is the number of negative cycles, zX is the total
number of cycles, and λ1≤…≤λνX, respectively λνX+1≤⋯≤λzX, are the lengths of the negative, respectively
positive, cycles of X. So any element of X has λ(X) as
its signed cycle type.
Our main aim in this section is to prove the first part of Theorem 1.2 along with the following result.
Theorem 3.1.
Suppose W^ is a Coxeter group of type
Dn, and let C^ be a conjugacy class of W^. Set C=C^0=C^w0, where w0 is the longest element of W, and assume
that λ(C)=(λ1,…,λνC;λν+1,…,λzC). Then the maximal length in C^
is
n2+zC-2∑i=1νC-1(νC-i)λi.
Let Φ be the root system of W. We employ the usual
description of Φ (as given, for example in [10]).
So the positive long roots are
Φlong+={ei±ej:1≤i<j≤n},
the negative long roots are
Φlong-=-Φlong+
and
Φlong=Φlong+∪Φlong-. The short roots are
Φshort+={ei:1≤i≤n},Φshort-=-Φshort+
and Φshort=Φshort+∪Φshort-. Finally, the positive roots are
Φ+=Φlong+∪Φshort+,
the negative roots are
Φ-=Φlong-∪Φshort-
and Φ=Φ+∪Φ-. We note that the
set of positive roots for W^ is Φlong+. We recall our convention will be
that the action of a group element is on the left of the root, so
that for example
(1-3+8+)(e1)=(1+3+8+)(1-)(e1)=-e3.
For w∈W, we define the following two sets:
Λ(w)={α∈Φlong+:w(α)∈Φ-},
Σ(w)={α∈Φshort+:w(α)∈Φ-}.
Set lB(w)=|Λ(w)|+|Σ(w)| and lD(w)=|Λ(w)|. By [10], lB(w) is the length of w
and, should w∈W^, then lD(w) is the length of w
viewed as an element of W^. We call lB(w) the B-length of
w and lD(w) the D-length of w. Given
w∈W, let w¯ be the corresponding element of
Sym(n). So, for example, if
w=(1-3+8+),
then
w¯=(138).
Observe that for w∈W, by a slight abuse of notation, we can write
w=w¯(∏ei∈Σ(w)(i-)).
Hence, in our above example, (1-3+8+)=(138)(1-).
Later when we talk about excess in these groups, to avoid ambiguity we will use the notation eB(w) to mean the excess e(w) when w is viewed as an element of W, and eD(w) to mean the excess e(w) when w is viewed (where appropriate) as an element of W^. That is, for all w in W we define
eB(w)=min{ℓB(σ)+ℓB(τ)-ℓB(w):σ,τ∈W,w=στ,σ2=τ2=1};
eD(w)=min{ℓD(σ)+ℓD(τ)-ℓD(w):σ,τ∈W^,w=στ,σ2=τ2=1}.
As noted earlier, conjugacy classes of W are
parameterized by signed cycle type. So,
for example, if W is of type B9 and C is the W-conjugacy
class of
w=(1+2+)(3+4-5+)(6-7+8+)(9+),
then the signed cycle type λ(C) of C is λ(C)=(3,3;1,2).
In W^, conjugacy classes are also parameterized by signed
cycle type, with the exception that there are two classes for each
signed cycle type consisting only of even length, positive cycles.
(The length profiles in each pair of split classes are identical, because the classes are interchanged by the length-preserving graph automorphism.)
Lemma 3.2.
Let C be a conjugacy class of W, and w∈C. Then
|Λ(w)|≥n-zC+2∑i=1νC-1(νC-i)λi.
Moreover, |Σ(w)|≥νC.
Proof.
Set ν=νC and z=zC. Write w as a product of disjoint cycles, that is, w=σ1σ2⋯σz, where σ1,…,σν are negative cycles and the remaining cycles are
positive. Also, order the negative cycles such that i<j if and
only if the minimal element in supp(σi¯) is
smaller than the minimal element in supp(σj¯). Our
approach is to consider certain 〈w〉-orbits of
roots.
Firstly, let σ be a positive k-cycle of w and consider
the orbits consisting of roots of the form ea-eb, for a,b∈supp(σ¯) and a≠b. Each such orbit has length k. There
are 2(k2) roots of this form, and hence k-1 such orbits.
Let c be the maximal element in supp(σ¯). Then each orbit contains
both ea-ec and ec-eb for some a,b∈supp(σ¯). Now
ea-ac∈Φ+ and ec-eb∈Φ-. Therefore each
orbit includes a transition from positive to negative (that is, a
positive root α for which w(α) is negative). Hence
each orbit contributes at least one root to Λ(w). Therefore
each positive k-cycle
contributes at least k-1 roots to Λ(w).
Next suppose σ is a negative k-cycle of w. This time we
consider orbits consisting of roots of the form ±ea±eb,
for a,b∈supp(σ¯) and a≠b. Each such orbit has length
2k. There are 4(k2) roots of this form, and hence
k-1 such orbits. Moreover, if α lies in one of these
orbits, then -α lies in the same orbit. Thus again each
orbit includes a transition from positive to negative and hence
contributes at least one root to Λ(w). Therefore each
negative k-cycle contributes at least k-1 roots to Λ(w).
Now suppose σi and σj are negative cycles, with i<j, and consider the union of all orbits consisting of roots of
the form ±ea±eb, where a∈supp(σi¯) and b∈supp(σj¯). Suppose |supp(σi¯)|=k and |supp(σj¯)|=ℓ. Let c
be minimal in supp(σi¯). Then every orbit contains some ±ec±eb for some b∈supp(σj¯). For every root of the form
ec±eb, we have
wk(ec±eb)=-ec±eb′ and w2k(ec±eb′)=ec±eb′′
for some b′,b′′∈supp(σj¯). Now ec±eb and ec±eb′′ are positive
roots, but -ec±eb′ is negative. Therefore in this orbit
or part of orbit there is at least one transition from positive to
negative. There are 2ℓ roots of the form ec±eb, and
hence each pair σi, σj of negative cycles with i<j contributes at least 2|supp(σj¯)| roots to Λ(w). For
example, letting i range from 1 to ν-1, we get a total of
(ν-1)×2|supp(σν¯)| roots from pairs σi
and σν.
Combining these three observations and writing ki for |supp(σi¯)|, we see that
Λ(w)≥∑i=1z(ki-1)+2∑i=2ν(i-1)ki.
Since {k1,…,kν}={λ1,…,λν}, and λ1≤λ2≤⋯≤λν, it is clear that
∑i=2ν(i-1)ki=k2+2k3+⋯+(ν-1)kν
≥λν-1+2λν-2+⋯+(ν-1)λ1
=∑i=1ν-1(ν-i)λi.
Therefore
|Λ(w)|≥n-z+2∑i=1ν-1(ν-i)λi.
It only remains to show that |Σ(w)|≥ν. This
trivially follows from the fact that there are ν negative
cycles and each negative cycle must contain at least one minus
sign. Therefore there are at least ν roots ea for which
w(ea)∈Φ-. Thus |Σ(w)|≥ν and the proof of
the lemma is complete.∎
Next, given a conjugacy class C of W we define a particular
element uC of C (which will turn out to have minimal
B-length). Recall that the signed cycle type of C is
λ(C)=(λ1,λ2,…,λνC;λνC+1,…,λzC),
and write μi=n-∑j=1iλj for 1≤i<zC. Set ν=νC
and z=zC. Then define uC to be the following element of C:
uC=(1+2+⋯λz+)(μz-1+1+⋯μz-2+)⋯(μν+1+1+μν+1+2+⋯μν+)⋅
⋅(μν+1+μν+2+⋯μν-1-1+μν-1-)⋯(μ1+1+⋯n-1+n-).
As an example, let
w=(1-7+2-9-)(3-4+6-)(5+8-)
and let C be the conjugacy class of w in
type B9. Then λC=(2,4;3), νC=2, zC=3,
μ1=7 and μ2=3. This gives
uC=(1+2+3+)(4+5+6+7-)(8+9-).
Lemma 3.3.
Suppose w=uC for some conjugacy class C of W. Then
|Σ(w)|=νC 𝑎𝑛𝑑 |Λ(w)|=n-zC+2∑i=1νC-1(νC-i)λi.
Proof.
Again set z=zC and ν=νC. The size of Σ(w) is simply the number of
minus signs appearing in the expression for w. Here, Σ(w)={en,eμ1,…,eμν-1} and |Σ(w)|=ν.
To find Λ(w), consider pairs (i,j) with 1≤i<j≤n. Suppose first that i and j are in the same cycle of
w¯. Then ei∉Σ(w) because only the maximal
element of each negative cycle has a minus sign above it. If j=μk for some k, or if j=n, then exactly one of ei+ej∈Λ(w) or ei-ej∈Λ(w) occurs (depending
whether k<ν). Otherwise, ei-ej∉Λ(w) and
ei+ej∉Λ(w). Hence a cycle
(μk+1+1+⋯μk-1+μk±)
contributes exactly λk+1-1 roots to Λ(w).
Now suppose that i and j are in different cycles. Hence
w¯(i)<w¯(j). It is a simple matter to
check that if ei∈Σ(w), then {ei+ej,ei-ej}⊆Λ(w), whereas if ei∉Σ(w), then
ei-ej and ei+ej are not in Λ(w). Therefore each
i with ei∈Σ(w) contributes exactly 2(n-i)
additional roots to Λ(w), and no roots are contributed
when ei∉Σ(w).
Therefore
|Λ(w)|=∑k=1z(λk+1-1)+∑k:ek∈Σ(w)2(n-k)
=(n-z)+2((n-n)+(n-μ1)+(n-μ2)+⋯+(n-μν-1))
=(n-z)+2∑i=1ν-1∑j=1iλj
=n-z+2∑i=1ν-1(ν-i)λi.
Therefore |Λ(w)|=n-zC+2∑i=1νC-1(νC-i)λi and |Σ(w)|=νC.
∎
Corollary 3.4.
Let C be a conjugacy class of W. Then the
minimal B-length in C is
n+νC-zC+2∑i=1νC-1(νC-i)λi.
If w∈C has
minimal B-length, then
|Λ(w)|=n-zC+2∑i=1νC-1(νC-i)λi 𝑎𝑛𝑑 |Σ(w)|=νC.
Moreover, uC is a representative of minimal B-length
in C.
In the next corollary the element uCt is the element obtained
from uC by taking its shortest positive cycle (which in this
context will be the cycle (n+n-1+⋯m+) for some odd
m), and putting minus signs over n and n-1. In other words
it is the conjugate of uC by t=(n-). Conjugation by (n-) is the length preserving automorphism of W^ induced by the graph automorphism of the Coxeter graph Dn.
Corollary 3.5.
Let C be a conjugacy class of W. If C is also a conjugacy class, or a union of conjugacy classes,
of W^, then the minimal D-length of elements in the
class(es) is
n-zC+2∑i=1νC-1(νC-i)λi.
Moreover, uC and uCt are representatives of minimal
D-length in the class(es), with one in each W^-class if
the class C splits.
Theorem 3.6.
Let C be a conjugacy class of W and w∈C.
Let C0 be the conjugacy class of ww0 where w0 is the
longest element of W. Then the maximal B-length of elements of
C is n2-|Λ(uC0)|-|Σ(uC0)|, with uC0w0 being an element of maximal B-length.
If C is a conjugacy class or union of conjugacy classes
of W^, the maximal D-length of elements of C is
n2-n-|Λ(uC0)|. Moreover, uC0w0 and uC0tw0 are representatives of
maximal D-length in the class(es), with one in each W^-class if C is a split class.
Proof.
Let C be a conjugacy class of W. Since w0
is central, the W-conjugacy class C0 of ww0 is just
Cw0. Moreover, for any root α we have w0(α)=-α. Hence for all x∈W, |Λ(xw0)|=(n2-n)-|Λ(x)| and |Σ(xw0)|=n-|Σ(x)|. (Note that
there are n2-n long positive roots and n short positive roots.)
Let u=uC0. Then by Lemma 3.2 and Lemma 3.3,
we have that for all v∈C0, |Λ(v)|≥|Λ(u)| and |Σ(v)|≥|Σ(u)|. Now every x∈C is of the form vw0 for some v∈C0. Hence for every x∈C, we have
|Λ(x)|≤n2-n-|Λ(u)| and |Σ(v)|≤n-|Σ(u)|.
Also |Λ(uw0)|=n2-n-|Λ(u)| and |Σ(uw0)|=n-|Σ(u)|. Therefore the maximal B-length in C is
n2-n-|Λ(u)|+n-|Σ(u)|=n2-|Λ(u)|-|Σ(u)|
and this is attained by the element uw0. Moreover,
if C is a conjugacy class (or union of conjugacy classes) of
W^, then the maximal D-length is n2-n-|Λ(u)|
and this is attained by uw0 (or (uw0)t if the class
splits).∎
Theorem 3.1 now follows immediately from Theorem 3.6 and
Lemma 3.3. All that remains in this section is to prove the following corollary, which is the first part of Theorem 1.2.
Corollary 3.7.
Every conjugacy class of W contains an element wλ,ρ, where λ is a maximal split partition with respect to ρ. Each element wλ,ρ has maximal B-length and maximal D-length in its conjugacy class of W.
Proof.
Note that each element wλ,ρ, where λ=(λ1,…,λm) is a maximal split partition with respect to ρ, is of the form w0uC for some uC. In particular, we have zC=m and νC=m-ρ. Thus each element wλ,ρ has maximal B-length and maximal D-length in the class Cw0. Therefore, given any class C′ of W, setting C=C′w0, we see that w0uC is wλ,ρ for some suitable λ,ρ, and so wλ,ρ is of maximal B-length and D-length in C′. ∎
It is the task of the next section to show that these elements wλ,ρ have excess zero.
4 Excess zero in types Bn and Dn
The aim of this section is to prove Theorems 1.1 and 1.2 for W and W^. In order to do this, we will show that the elements wλ,ρ described in Theorem 1.2 have excess zero both in W and (if applicable) in W^. To obtain the required involutions σ and τ such that N(σ)∩N(τ)=∅ and στ=w, we modify the definition of gb1,…,bk given in Section 2. We will only need to consider sequences of consecutive integers here though. Let {a+1,a+2,…,a+k} be a sequence of consecutive positive integers in {1,…,n}. Define ga;k to be the permutation of W that reverses the sequence and fixes all other c∈{1,…,n}. (Essentially this is just gb1,…,bk, where b1=a+1, b2=a+2,…,bk=a+k, but viewed as an element of W rather than Sym(n).) Thus
ga;k(a+i)=a+k+1-i for 1≤i≤k. That is,
ga;k=(a+1+a+k+)(a+2+a+k-1+)⋯(a+⌊k/2⌋+a+⌈k/2⌉+1+).
In particular, ga;k is an involution.
We also define ha;k to be ga;k with the plus signs replaced by minus signs. Thus ha;k(a+i)=-(a+k+1)+i for 1≤i≤k. So
ha;k={(a+1-a+k-)(a+2-a+k-1-)⋯(a+k2-a+k2+1-),k even,(a+1-a+k-)(a+2-a+k-1-)⋯(a+⌊k2⌋-a+⌈k2⌉+1-)(a+⌈k2⌉-),k odd.
In particular, ha;k is an involution. Moreover, ha;k is order preserving on the intervals [1,a], [a+1,a+k] and [a+k+1,n].
As an example
g1;6=(2+7+)(3+6+)(4+5+) and h3;5=(4-8-)(5-7-)(6-).
Next we define two kinds of cycle and some involutions which are relevant to our analysis of the elements wλ,ρ. Define
wa;k-=(a+1-a+2-⋯a+k-1-a+k-),
σ(wa;k-)=ha;k,
τ(wa;k-)=ga;k-1,
wa;k+=(a+1-a+2-⋯a+k-1-a+k+),
σ(wa;k+)=ha+1;k-1,
τ(wa;k+)=ga;k.
Lemma 4.1.
Let w be either wa;k- or wa;k+. Then writing σ=σ(w) and τ=τ(w), we have that w=στ, where σ and τ are both involutions.
Proof.
It is clear from the definitions that σ and τ are involutions. First we consider w=wa;k-. Then if 1≤i≤k-1, we have
στ(a+i)=σ(a+k-i)=-(a+k+1)+(k-i)=-(a+i+1),
whereas
στ(a+k)=σ(a+k)=-(a+k+1)+k=-(a+1).
Therefore w=στ in this case.
Now consider w=wa;k+. Then if 1≤i≤k-1, we have
στ(a+i)=σ(a+k+1-i)=σ((a+1)+(k-i))=-((a+1)+(k-1)+1)+(k-i)=-(a+i+1),
whereas
στ(a+k)=σ(a+1)=a+1.
Thus again w=στ and the proof is complete.∎
Proposition 4.2.
Let w=wλ,ρ be the corresponding signed element of the maximal split partition λ=(λ1,…,λm) (with respect to ρ). Then
eB(w)=eD(w)=0.
Proof.
By definition, and recalling that μi=∑j=1i-1λj, we have w=w1⋯wm, where
wi={(μi+1-μi+2-⋯μi+1-1-μi+1-)if 1≤i≤ρ,(μi+1-μi+2-⋯μi+1-1-μi+1+)if ρ<i≤m.
Therefore
wi={wμi;λi-if 1≤i≤ρ,wμi;λi+if ρ<i≤m.
For each i set σi=σ(wi) and τi=τ(wi). Since the supports of σi¯ and τi¯ are subsets of the support of wi, it is clear that both σi and τi commute with both σj and τj whenever i≠j. Hence σ=σ1⋯σm and τ=τ1⋯τm are involutions with the property that στ=w.
We must show that N(σ)∩N(τ)=∅.
Consider a cycle wk of w. Then τ(wk) is either gμk;λk-1 or gμk;λk. The action of g is to reverse the order of the sequence μk+1,…,μk+λk, reverse the order of the sequence -μk,…,-μk-λk and fix all other integers. Hence
(4.1)N(τ(wk))⊆{ei-ej:μk<i<j≤μk+1}.
On the other hand σ(wk) is either hμk;λk or hμk+1;λk-1, so is of the form ha;b, where a≥μk and a+b=μk+1.
We observe that
(4.2)Σ(ha;b)={ea+1,…,ea+b}⊆{eμk+1,…,eμk+1}.
Recall that ha;b fixes ei for all i∉{a+1,…,a+b} and ha;b(ei)=-e2a+b+1-i if i∈{a+1,…,a+b}. From this we see that
Λ(ha;b)={ei+ej:a<i<j≤a+b}
∪{ei±ej:a<i<a+b<j≤n}.
Therefore
(4.3)Λ(σ(wk))⊆{ei+ej:μk<i<j≤μk+1}∪{ei±ej:μk<i<μk+1<j≤n}.
For l≠k, we note that σl and τl fix all i for i∉{μl+1,…,μl+1}. In particular, they stabilise (setwise) the sets {1,…,μk}, {μk+1,…,μk+1} and {μk+1+1,…,n}. Therefore σl(N(σk))=N(σk). So we may apply Lemma 1.4 to conclude that
N(σ)=⋃˙i=1mN(σk)
and that
N(τ)=⋃˙i=1mN(τk).
Equations (4.1), (4.2) and (4.3) now imply that N(τ)∩N(σ)=∅. Therefore by equation (1.1) we see that ℓB(w)=ℓB(σ)+ℓB(τ) and therefore eB(w)=0. But also N(τ)∩N(σ)=∅ implies that Λ(τ)∩Λ(σ)=∅, and so we also have ℓD(w)=ℓD(σ)+ℓD(τ), giving eD(w)=0 as required.
∎
We observe that Theorem 1.2 now follows immediately from Corollary 3.7 and Proposition 4.2.
Corollary 4.3.
Theorem 1.1 holds for Coxeter groups of type Bn and Dn.
Proof.
If W is of type Bn, then by Theorem 1.2 every conjugacy class C of W contains an element of the form wλ,ρ for suitable λ and ρ, and this element has excess zero and maximal B-length in C. Now consider W^ of type Dn, and let C be a conjugacy class of W^. If C is also a conjugacy class of W, then again C contains some wλ,ρ, which has maximal D-length and excess zero. If C is not a conjugacy class of W, then C∪C(n-) is a conjugacy class of W (as conjugation by (n-) is a length preserving map corresponding to the non-trivial graph automorphism of Dn), so for some w=wλ,ρ we have either w or w(n-)∈C. Now e(w)=0, which means there are σ,τ involutions such that w=στ and ℓ(w)=ℓ(σ)+ℓ(τ). Hence
w(n-)=σ(n-)τ(n-)
and, since conjugation by (n-) is a length-preserving map, we have
ℓ(w(n-))=ℓ(σ(n-))+ℓ(τ(n-)).
Hence either w or w(n-) is an element of maximal D-length and excess zero in C.
∎
5 Conclusion
Proof of Theorem 1.1.
Observe that every finite Coxeter group W is a direct product of irreducible Coxeter groups. If W=W1×⋯×Wn for some Wi, then it is easy to see that for w=(w1,…,wn)∈W, we have
ℓ(w)=ℓ(w1)+⋯+ℓ(wn)
and
e(w)=e(w1)+⋯+e(wn).
Moreover, w is of maximal length in some conjugacy class C of W if and only if each wi is of maximal length in a conjugacy class of Wi. Therefore Theorem 1.1 holds if and only if it holds for all finite irreducible Coxeter groups. Theorem 1.1 has already been proved for types An, Bn and Dn (Proposition 2.4 and Corollary 4.3). The exceptional groups E6, E7, E8, F4, H3 and H4 were checked using the computer algebra package Magma [1]. In each case there is at least one (usually many) elements of maximal length and excess zero in every conjugacy class. Finally, it is easy to check that every element of the dihedral group has excess zero, so the result is trivially true. Thus Theorem 1.1 holds for every finite irreducible Coxeter group, and hence for all finite Coxeter groups.
∎
Theorem 1.1 shows the existence of at least one element of maximal length and excess zero in every conjugacy class of a finite Coxeter group. However, if one looks at some small examples in the classical Weyl groups, it appears that every element of maximal length in a conjugacy class has excess zero. It is natural to ask whether this holds in general. It turns out that it does not – although the number of elements for which it fails seems to be small. For example, if W is of type E6, then in 23 of the 25 conjugacy classes every element of maximal length has excess zero. If W is of type E7, then every element of maximal length in 59 of the 60 conjugacy classes has excess zero. In the remaining class, which consists of elements of order 3, there are 708 elements of maximal length, all but 50 of which have excess zero.
For the classical Weyl groups, we have checked all conjugacy classes of these groups for n up to 10, and in each case every element of maximal length in a conjugacy class has excess zero. However, as Lemma 5.1 shows, there are examples of elements w of maximal length with an arbitrarily large number of pairs of involutions xy with w=xy, such that only one such pair has the property that ℓ(w)=ℓ(x)+ℓ(y). Elements like these “only just” have zero excess. Because of near misses such as this, we are not sufficiently confident that the pattern of maximal length elements having zero excess will continue, even in classical groups. It would be interesting to know whether it does.
Lemma 5.1.
Let W be of type Bn, for n≥2. There are at least 2n pairs of involutions (x,y) such that
xy=(1-2+),
but only one of these pairs has the property that ℓ(x)+ℓ(y)=ℓ((1-2+)).
Proof.
The element w=(1-2+) is certainly of maximal length in its conjugacy class, by Theorem 1.2.
If x is an involution such that xy=w for some involution y, then wx=w-1. Thus x=x1x2, where x1 and x2 are commuting involutions, x2 fixes 1 and 2, and x1 is either
(1-),(2-),(1-2-) or (1+2+).
We can then determine y, and the upshot is that we get the following possibilities, where here z is any involution fixing 1 and 2.
| x | y |
| (1-)z | (1-2-)z |
| (2-)z | (1+2+)z |
| (1-2-)z | (2-)z |
| (1+2+)z | (1-)z |
If N(x)∩N(y)=∅, then clearly we must have z=1. It is now a quick check to show that the only possibility is
x=(2-),y=(1+2+).
The number of possible pairs (x,y) is four times the number of involutions in a Coxeter group of type Bn-2, which is
at least 2n-2, because for all subsets {a1,…,ak} of size k of {3,4,…,n} the element (a-1)⋯(a-k) is an involution.
∎
and
Peter J. Rowley
